Dragons are powerful and intelligent serpentine creatures from across the multiverse. They are huge, flying reptiles with formidable physical strength, and many of them also score high on intelligence and magical capabilities. They can be found in all facets of the Color Pie, although more dragons are aligned with red than with any other color. Dragon is the iconic creature for the color red, and at least one red Dragon has appeared in every core set.  
Baby or miniature dragons are referred to as "hatchlings" and "whelps".
The Elder Dragons were ancient dragons of Dominia. Their vast power rivaled that of planeswalkers. The chronologically oldest bit of Magic lore we have is that about the Elder Dragon Wars, a conflict that shook the planes and killed most of their kind. The winners of the war became the ancestors of all dragons and drakes, while the losers were stripped of their wings, limbs and most of their power and became the Elder Land Wurms, which spawned all other wurm species. Viashino are also descendants of them, although it is unknown how.
There is some debate about how many Elder Dragons actually survived. We know for sure that the siblings Nicol Bolas, Chromium Rhuell, Arcades Sabboth, Palladia-Mors, and their cousin Vaevictis Asmadi survived, but were there any others? Some prerevisionist lore says there were none, and that Piru was a Lesser Elder, but it seems unlikely that a race as powerful as the dragons could arise from five close relatives. It could be that the five Elder Dragon Legends were simply the most prominent of the survivors. Presently, Nicol Bolas is the only known Elder Dragon that still lives and is the most powerful of the Elder Dragons.
Also known as Lower Dragons. Their name suggests that they are insignificant or weak, but this is only the case when compared to the Elder Dragons. The Lesser Elders where the firstborn children of the Elders. They were less powerful than their parents but still very mighty and they sired the many races of dragons we know today.
Known Lesser Elders are Scarzam dragons, and possibly Piru.
The Scarzam dragons were a race of dragons sired by the Lesser Elder Scarzam and used by a woman, Sivitri Scarzam, to wage war on Corondor. They were nearly wiped out by a magical, poisonous flower created by Sol'Kanar to destroy Sivitri's power. Only one, and the biggest of these dragons, survived.
The five Primeval Dragons were self-proclaimed gods and once ruled Dominaria until the Numena managed to imprison them and steal their power. The Primevals were ordinary dragons until all of them were awakened, when their combined magic unlocked their magnificent powers.
During the Phyrexian Invasion, Tevesh Szat revealed the existence of the Primevals to Rhammidarigaaz, also known as Darigaaz, who was in fact a reincarnated Primeval himself. After they were all reawakened, they went on a rampage against both Dominarians and Phyrexians, until Karn convinced Darigaaz of his folly. Darigaaz committed suicide by plunging into a volcano after which the other Primevals could be defeated.
Their names had a special meaning in Old Draconic describing the stages of a dragon's life. The first Primeval was Rhammidarigaaz ("conception"), the second was Rith ("childhood"), the third was Treva ("youth"), the fourth was Dromar ("adulthood"), and the last was Crosis ("death").
In addition to being godlike themselves, each dragon has a characteristic associated with belonging to the Ur-Dragon, the spirit of all dragonkind. Each characteristic can be found in the flavor text of its respective attendant golem.
Draco is a powerful artifact dragon. It is not known what Draco is or originally was, but the artwork suggests that it might had been a natural dragon (Instead of a pure artificial construct) that was later modified with certain metallic parts. other artifact dragons are Tek and the Clockwork Dragon  Teeka's dragon was an artificial dragon created by the artificer Teeka in the short story "Better Mousetrap" in the Distant Planes anthology.
Powerful Hellkites known as Hellkite Overlords inhabit the shard of Jund
The Krosan dragons of Otaria have no wings but have six legs, are covered in thorny armor and breathe a magical vapor that causes wild vegetation to grow in their presence. Smaller Krosan dragons called dragonettes were used in the Cabal's pits. However, this caused a fully-grown dragon to come to Cabal City to try to avenge its kin.
Nekoru, or Cat-Dragons, are cat-like beings with reptilian wings from Jamuraa. It is unknown whether they where actually descendants from the Elder Dragons, although their non-reptilian bodies seem to suggest otherwise.
The nekoru Wasitora played an important role in the second Legends Cycle. The race was revisited in Time Spiral — when Teferi's host is transported to Madara near Talon Gates thanks to the call of Nicol Bolas, the party meets the nekoru prince Yurei-teki.
The name "Nekoru" comes from the Japanese word "Neko", for "cat", and "ru" (or "ryu"), for "dragon". Literally translated it means "cat-dragon".
Shivan dragons were the most powerful red-aligned dragons and hailed from the island of Shiv. Sometimes they allied themselves with the Viashino of their homeland; sometimes they fought against them. After Teferi phased out a large chunk of Shiv, some of these dragons tried to rebuild their native land, many journeyed away to seek any opportunity to restore the broken pride of their race; Rorix Bladewing was one of those dragons. Rorix fought and died in the Grand Coliseum, but after his death he was reanimated as Bladewing the Risen.
Viashivan dragons were Jamuraan dragons rivaling the Shivan dragons in power.
Archwing dragons are reputed to be swifter than an angel, crueler than a demon, and relentless as a ghoul. Balefire dragons are a threat to the human population while Moonveil dragons are considered allies.
The furnace dragons lived on Mirrodin.  They were probably brought there by Memnarch's soul traps and parts of their bodies were turned into metal by the mycosynth that grows inside Mirrodin. The metal components augmented their offensive power and mobility. Furnace dragons had a literal furnace inside their chest, which generated both their breath weapon and bursts of fire from "exhaust vents" in their wings. Unlike most other dragons, they didn't glide on wind currents; they flew through Mirrodin's skies on pure explosive power. Their metallic modifications left them inherently unstable. Overloading their furnaces for particularly large periods of flame-production could have explosive repercussions. A known (although possibly mythic) furnace dragon is Chiss-Goria.
New Phyrexian dragons
The dragon spirits are powerful kami of Kamigawa. There are five, one for each color of magic. During the Spirit War, they did not side with O-Kagachi, but remained true to their pledge to protect several important places of the mortals.
- Yosei, the Morning Star protected Eiganjo. He was ripped apart in the battle against O-Kagachi, but survived and helped the Eiganjoan refugees in the battle against the hordes of the Myojin of Infinite Rage.
- Keiga, the Tide Star protected Minamo. She was killed (if kami can be killed) by Hidetsugu and his yamabushi minions during their attack on the school.
- Jugan, the Rising Star protected the forest of Jukai. In the initial stages of the war, he was summoned by the monks of the Order of the Sacred Bell to fight in the Battle of Silk. He was killed, but blessed the monks with power even after he was gone. During the final stages of the war he was summoned by Sosuke and Sachi to aid the Orochi against the Soratami invaders. Apparently he had the ability to come back from the dead (perhaps all kami can).
- Kokusho, the Evening Star was the guardian of the Takenuma swamp.
- Ryusei, the Falling Star was the guardian of the Sokenzan mountains.
The Kamigawa vignette on Kiki-Jiki reveals that there are more dragons on Kamigawa and that not all of them sided against O-Kagachi. Nothing more is known about them though.
It is important to note that while they have the shape of a dragon, they are in fact spirits foremost. This is an important distinction when considering the planeswalker Ugin who is known as a spirit dragon and therefore does not have as much in common with them as one would assume.
Dragon engines were dragon-like creatures of unknown origin that lived in the first sphere of Phyrexia. They were used by Mishra during the Brothers' War (he used real Phyrexian dragon engines and his own versions which were inferior) and by Phyrexia itself in the Invasion.
Remains of them can still be found in Dominaria among the wreckage of the Invasion.
On the plane of Ravnica, with dragons deemed too dangerous to exist, the Guildpact was signed in part to rid the world of such beings; around the Decamillennial almost all of them were dead. The most significant exception being Niv-Mizzet, the parun and Guildleader of the Izzet League.
Dragon tokens are produced by:
- Brood Keeper (2/2 red Dragon with Flying)
- Broodmate Dragon (4/4 red Dragon with Flying)
- Day of the Dragons (5/5 red Dragon with Flying)
- Death by Dragons (5/5 red Dragon with Flying)
- Dragon Broodmother (1/1 red and green Dragon with Flying and Devour:2)
- Dragon Egg (2/2 red Dragon with Flying)
- Dragon Roost (5/5 red Dragon with Flying) 
- Dragonmaster Outcast (5/5 red Dragon with Flying)
- Sarkhan Vol (4/4 red Dragon with Flying)
- Sarkhan the Mad (5/5 red Dragon with Flying)
- Tatsumasa, the Dragon's Fang (5/5 blue Dragon Spirit with Flying)
- Utvara Hellkite (6/6 red Dragon with flying)
- Dragon Engine isn't a dragon, it is a construct.
- Nalathni Dragon is the smallest dragon ever printed and is the only dragon with banding. 
- The lord-like enchantment Crucible of Fire grants +3/+3 to dragons you control.
- Dragonshift changes creatures into dragons.
- The Shichifukujin Dragon is one of only two unique Magic cards in existence (meaning there's only one copy in existence). 
- Catacomb Dragon and crux of fate(yet to be printed in fate reforged) are the only cards that refers to non-dragons. 
In other languages
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- Magic Arcana. (March 16, 2004.) "Artifact dragons", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Magic Arcana. (April 27, 2004.) "Dragons on Echoing Truth", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Magic Arcana. (April 29, 2003.) "Non-Dragon Dragon cards", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Magic Arcana. (April 28, 2003.) "Nalathni Dragon", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Mark Rosewater. (February 14, 2005.) "Land of the Rising Fun", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Magic Arcana. (April 10, 2006.) ""Non-X" Beware!", Daily MTG, magicthegathering.com, Wizards of the Coast.